#139681

order	512
nilpotency class	4
isoclinism	isoclinic class of #139568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#416
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

294 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 80 points:

(1,2,5,9,6,3,7,11)(4,10,13,16,12,8,14,15)(17,18)(19,29)(20,25,36,49)(21,38)(22,43)(23,27)(24,32)(26,41)(28,40)(30,52)(31,59,61,53)(33,56)(34,67)(35,68,39,71)(37,70,69,50)(42,44)(45,63,73,60)(46,57)(47,65)(48,72,51,58)(54,66)(55,64)(62,75,79,78)(74,77,76,80);(1,3,5,11,6,2,7,9)(4,10,14,15,12,8,13,16)(17,19,30,57,22,34,21,32,23,33,41,66,40,64,42,65)(18,24,44,67,28,46,26,29,27,47,38,55,43,54,52,56)(20,31,58,77,37,62,39,63,36,61,72,80,69,79,35,60)(25,45,68,78,50,74,51,53,49,73,71,75,70,76,48,59);(1,4)(2,8)(3,10)(5,14)(6,12)(7,13)(9,16)(11,15)(17,20,22,37,23,36,40,69)(18,25,28,50,27,49,43,70)(19,31,34,62,33,61,64,79)(21,39,41,72,42,35,30,58)(24,45,46,74,47,73,54,76)(26,51,38,71,52,48,44,68)(29,53,55,75,56,59,67,78)(32,63,66,80,65,60,57,77)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=g,d^2=gi,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,e^b=eh,d^c=dh,e^c=ei,f^c=fh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46877);